Find each product. See Examples 5 and 6. (2z^4-3y)^2

Verified step by step guidance

1. Recognize that the expression \((2z^4 - 3y)^2\) is a binomial squared, which can be expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify \(a = 2z^4\) and \(b = 3y\) in the expression \((2z^4 - 3y)^2\).
3. Calculate \(a^2\) by squaring \(2z^4\), which results in \((2z^4)^2 = 4z^8\).
4. Calculate \(2ab\) by multiplying \(2\), \(a = 2z^4\), and \(b = 3y\), which results in \(2 \times 2z^4 \times 3y = 12z^4y\).
5. Calculate \(b^2\) by squaring \(3y\), which results in \((3y)^2 = 9y^2\).

Key Concepts

Polynomial Expansion

Polynomial expansion involves multiplying out expressions to express them in a standard form. In this case, the expression (2z^4 - 3y)^2 requires applying the distributive property or the binomial theorem to expand the square of a binomial. This process results in a polynomial that combines like terms.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem simplifies the process of expansion, especially for higher powers, by systematically calculating each term.

Combining Like Terms

Combining like terms is a fundamental algebraic process where terms with the same variable and exponent are added or subtracted. After expanding a polynomial, it is essential to identify and combine these terms to simplify the expression into its most concise form, ensuring clarity and ease of further calculations.